Problem: All of the 5th grade teachers and students from Springer went on a field trip to an art museum. Tickets were $$6.00$ each for teachers and $$3.50$ each for students, and the group paid $$47.00$ in total. A few weeks later, the same group visited a science museum where the tickets cost $$24.00$ each for teachers and $$9.50$ each for students, and the group paid $$143.00$ in total. Find the number of teachers and students on the field trips.
Solution: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${6x+3.5y = 47}$ ${24x+9.5y = 143}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-4$ ${-24x-14y = -188}$ ${24x+9.5y = 143}$ Add the top and bottom equations together. $ -4.5y = -45 $ $ y = \dfrac{-45}{-4.5}$ ${y = 10}$ Now that you know ${y = 10}$ , plug it back into $ {6x+3.5y = 47}$ to find $x$ ${6x + 3.5}{(10)}{= 47}$ $6x+35 = 47$ $6x = 12$ $x = \dfrac{12}{6}$ ${x = 2}$ You can also plug ${y = 10}$ into $ {24x+9.5y = 143}$ and get the same answer for $x$ ${24x + 9.5}{(10)}{= 143}$ ${x = 2}$ There were $2$ teachers and $10$ students on the field trips.